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Can Isay,

(a) His mother put some ointment to ease his pain. He felt better / fine. Later, he continued his work. He promised to becareful next time.

(b) He promised (that) he would be (more) careful in the future / next time.
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Hi,

(a) His mother put on some ointment to ease his pain. He felt better / fine. Later, he continued his work. He promised to be careful next time.

(b) He promised (that) he would be (more) careful in the future / next time.

This is all OK.

in the future suggests every time in the future.

next time often means the same as above, but can also just mean one time in the future.

Best wishes, Clive
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Can I say,

(a) He promised to his mother be careful next time.

(b) He promised his mother to be careful next time.
Hi,

Only (b) is correct.

Clive
How about this sentence?

(a) He promised next time to be careful .

(b) He promised next time he would be careful.

(c) He promised his mother would be careful next time.
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Hi,

How about this sentence?

(a) He promised to be careful next time .

(b) He promised he would be careful next time.

(c) He promised his mother he would be careful next time.

Best wishes, Clive